Sampling calculator: match your camera and telescope to the seeing

What sampling means

A star in a long exposure is not a point. The atmosphere, the optics and the guiding spread it into a blob whose width at half its peak brightness is the FWHM (full width at half maximum), measured in arcseconds. Sampling asks how many pixels that width covers:

pixels per FWHM = star FWHM (″) ÷ image scale (″ per pixel)

Too few pixels and stars look square, and fine detail is lost between pixels (under-sampling). Too many and the extra pixels add no detail while each one collects less light, so per-pixel noise rises and files grow (over-sampling).

The rules of thumb, and why they differ

There is no single agreed number. The calculator offers three bands taken from published guidance:

Band	Source	Context
2–3 px per FWHM	AAVSO CCD/CMOS photometry guide	photometry; below 2 is under-sampled
2.5–3 px	Diffraction Limited	2 px minimum, 2.5–2.8 generally optimal for square pixels
3.3–4 px	Stan Moore	3.33 px as the critical value once the pixel diagonal is counted

The usual explanation links 2 pixels per FWHM to the Nyquist sampling theorem. A refereed paper on spectrograph sampling, Robertson (2017), argues that the “2 pixels = Nyquist” maxim is incorrect for typical profiles, which alias at that rate. Atik takes a different route and expresses its rule of thumb as 1–2″ per pixel for typical seeing, rather than a pixel count. The default band in the calculator is 2–3 pixels; switch bands to see how the verdict changes.

What FWHM to enter

The best input is the FWHM of stars measured in your own sub-exposures, if your capture or stacking software reports it. That measurement includes everything that blurs the stars: seeing, focus, optics, guiding. Without it, use typical values. For amateur sites, the AAVSO guide gives 3–4″ for most locations and Diffraction Limited 2.5–3.5″ for non-alpine sites. Professional observatories are in a different class: ESO reports a median seeing of 0.72″ at Paranal.

Worked examples

Assuming 2.5″ stars and the 2–3 pixel band:

Setup	Image scale	Pixels per FWHM	Result
3.76 µm pixels at 400 mm	1.94″	1.29	under-sampled
3.76 µm at 900 mm	0.86″	2.90	within
3.76 µm at 1280 mm	0.61″	4.13	over; 2×2 binning gives 2.06
3.76 µm at 2032 mm	0.38″	6.55	over; 3×3 binning gives 2.18
2.9 µm at 2032 mm	0.29″	8.49	over; 3×3 binning gives 2.83

For 2.5″ seeing and 3.76 µm pixels, the band corresponds to focal lengths of about 620–930 mm. With 3.5″ seeing it moves down to about 440–665 mm. The calculator prints this range for your own pixel size and FWHM.

What to do about it

• Over-sampled: bin 2×2 or 3×3 (the calculator lists each level), or use a reducer to shorten the focal length. The reducer calculator shows the new image scale.
• Under-sampled: wide-field imaging is often under-sampled by design, because field of view matters more than fine detail. Dithering between frames and drizzle integration can recover some resolution; the drizzle guide explains the method and its costs.
• Colour cameras: the image scale is per photosite, but each colour is sampled by only part of the Bayer pattern, so a one-shot colour camera samples each colour more coarsely than the numbers suggest. See the mono vs colour guide.

The sampling and seeing guide goes further into how FWHM is measured and where the competing rules come from.